Explore Newton's method for approximating cube roots in this 30-minute mathematics lecture. Delve into the concept of cubes and cube roots, examining why finding the exact cube root of 5 is impossible. Learn how to apply Newton's method to find numbers with cubes approximately equal to 5, and discover a higher variant using tangent conics instead of tangent lines. Investigate the similarities between this problem and finding square roots of 2 and 5. Gain insights into Taylor expansion, tangent lines, and the weighted Babylonian method. Conclude with an introduction to the Quadratic Newton Method and a practical exercise to reinforce your understanding of these advanced mathematical concepts.